This core framework can be further extended to the automatic discovery of PDEs by including partial derivatives in the candidate library [1]}, [2]}. Concretely, spatiotemporal data of \(m\)  spatial locations measured over \(n\)  time slices is arranged into a length \(mn\)  column vector \({\bf {U}}\) . Then a candidate library \(\Theta ({\bf U}) \in \mathbb {R}^{mn \times D}\)  is constructed as before except here we consider functions of both the system state and system spatial derivatives (which have to be numerically approximated). That is, in addition to library functions like \(\mathbf {U}^2\) , we also might include \(\mathbf {U}\mathbf {U}_x\) , and potentially higher order derivatives like \(\mathbf {U}_{xxx}\) . Finally, as before, we seek coefficients \({\Xi }\)  which accurately reconstruct the temporal dynamics \(\mathbf {U}_t = \Theta (\mathbf {U}) {\Xi }\) .
